# Residual Stress Distribution Design for Gear Surfaces Based on Genetic Algorithm Optimization

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Residual Stress Distribution

- (1)
- σ
_{surface}—the RS at the surface; - (2)
- σ
_{max}—the peak RS; - (3)
- y
_{max}—the depth of σ_{max}; - (4)
- y
_{core}—the depth where RS vanishes.

## 3. Gear Contact Model

^{*}in plane strain condition [23], as shown in Figure 3b.

_{rated}= 423 N/mm, which can be easily obtained using the parameters in Table 1 [27], is the normal contact load. To gain basic physical insights, the variation of R is ignored and R = 15.231 mm at the pitch point is used for the rigid cylinder. Due to the involute geometry, the gear surface is subjected to rolling–sliding contact. This type of contact loading can be represented by Hertzian pressure with shear traction:

_{c}is the x-coordinate of the normal pressure center, b = (4PR/(πE

^{*}))

^{0.5}is the half contact width, t is the shear traction, and μ is the friction coefficient ranging from 0 to 0.3, representing ideally lubricated to dry friction conditions.

_{c}is varied from −30b to 30b so that the pressure profile moves over a sufficiently long distance to ensure that the material points at x = 0 undergo a complete loading cycle.

## 4. Fatemi–Socie Multiaxial Fatigue Criterion

_{n,max}are the maximum shear strain range and maximum normal stress on this plane over one loading cycle, respectively; Y is the material yield strength, and k is a fitting parameter ranging from 0 to 1 [31]. For high cycle fatigue, which is the case in the present study, k approaches unity. Thus, k = 1 is adopted. The normal stress component is included to consider the fact that compressive stress tends to retard fatigue microcrack initiation while tensile stress accelerates the initiation. The maximum FSDP on all the critical planes is denoted by D

_{FS}. The fatigue initiation life N

_{f}of a material point is related to D

_{FS}by:

_{f}is the number of loading cycles when the fatigue microcrack is initiated; G = E/(2 + 2ν); ${\tau}_{\mathrm{f}}^{\prime}$ and ${\gamma}_{\mathrm{f}}^{\prime}$ are the shear fatigue strength coefficient and the shear fatigue ductility coefficient, respectively; m and n are the fatigue strength exponent and the fatigue ductility exponent, respectively. ${\tau}_{\mathrm{f}}^{\prime}$ = 1296 MPa, ${\gamma}_{\mathrm{f}}^{\prime}$ = 0.437, m = −0.087, and n = −0.58. Note that temperature may have an impact on the above parameters. However, this is not considered since this paper only focuses on demonstrating the methodology for the design of RS distribution. One can easily incorporate a temperature-dependent fatigue citation in the proposed methodology to obtain a temperature-dependent optimum RS distribution.

_{FS}and N

_{f}for one material point. The critical plane needs to be first identified. To this end, the stress–strain history over a loading cycle for a material point should be obtained. This can be done by taking the coordinates of this point into Equations (A1)–(A7) and varying x

_{c}from −30b to 30b. Then, all the candidate planes that pass this point are examined to find the critical plane. Each candidate plane is characterized by the angle α between its normal direction and the x-axis (see Figure 4b), ranging from 0° to 180° with intervals of 0.2°. The stress–strain components on each candidate plane can be obtained by coordinate transformation.

_{n,max}can thus be obtained. Then, the critical planes can be determined. D

_{FS}is the largest FSDP among all the critical planes. Finally, the fatigue initiation life of this point, N

_{f}, can be calculated using Equation (6).

## 5. Optimization Scheme

_{f})

_{min}, i.e., the minimum N

_{f}among all the points. Since (N

_{f})

_{min}increases with decreasing (D

_{FS})

_{max}according to Equation (6), the objective of this optimization problem is to minimize (D

_{FS})

_{max}. As can be seen from Figure 2, σ

_{surface}, σ

_{max}, y

_{max}, and y

_{core}are the four design variables for the optimization, and thus, the constraints are applied on these four variables.

_{surface}, σ

_{max}, y

_{max}, y

_{core}) is called an individual in GA. Each component of the individual is called a “gene”. The constraints in Equation (8b–g) are summarized from the existing experimental [38,39,40,41] and theoretical [13,42] results for RS distribution profiles produced using the shot peening process.

_{FS})

_{max}for each individual and use 1/(D

_{FS})

_{max}to measure its fitness. Individuals having higher fitness are more likely to survive.

_{c}, decides how many individuals are generated in the next generation. Finally, genes of individuals are altered randomly using the mutation operator.

- (1)
- The generation number reaches the prescribed upper limit G
_{max}; - (2)
- The relative change in the highest fitness over G
_{s}generations is less than the function tolerance value e.

## 6. Results and Discussion

#### 6.1. Mechanism for RS to Increase (N_{f})_{min}

_{n,max}, and the FSDP on all the candidate planes (0° ≤ α ≤ 180°) of material points at x = 0 for three working conditions where P/P

_{rated}= 1 with μ = 0, 0.1, and 0.3 without the presence of RS. These results are normalized by their maximum absolute values. The critical planes of each material point along the depth are denoted by the solid cyan lines in the Δγ and FSDP distribution.

_{n,max}on the fatigue damage is demonstrated by the difference between the distributions of Δγ and FSDP. σ

_{n,max}is compressive for μ = 0 and tensile for μ ≠ 0. For μ = 0 and 0.1, the effect of σ

_{n,max}is negligible. Due to the mild friction on the surface, the maximum damage occurs below the surface. On the other hand, for μ = 0.3, the effect of σ

_{n,max}is appreciable and the high friction gives rise to the maximum damage appearing on the surface. For all the three cases (P/P

_{rated}= 1; μ = 0, 0.1, and 0.3), since the σ

_{n,max}distribution over α is not uniform, the FSDP on the two critical planes is different according to Equation (5). Thus, there is only one critical plane having the largest FSDP (i.e., D

_{FS}) and the orientation of the critical plane corresponding to D

_{FS}is denoted by α

_{c}.

_{n,r}introduced by this RS distribution on all candidate planes. The σ

_{n,r}distribution in Figure 7b can be added to the σ

_{n,max}distribution without RS in Figure 6 to obtain the new σ

_{n,max}distribution with RS. In this way, the FSDP distribution can be changed by RS. Nonetheless, there is an exception that when α

_{c}is close to 90° (see Figure 6, μ = 0), N

_{f}will remain unchanged since RS will have zero normal components on the α

_{c}plane and will not be able to decrease D

_{FS}in such a case.

_{FS}distributions along the depth for different μ under the rated normal load. It is of interest to decrease (D

_{FS})

_{max}so that (N

_{f})

_{min}can be increased. From Figure 8a, as μ increases, the location of (D

_{FS})

_{max}moves from the depth of around 0.13 mm to the surface. μ

_{c}can be defined as the critical friction coefficient. For μ < μ

_{c}, the location of (D

_{FS})

_{max}is below the surface. For μ > μ

_{c}, the location of (D

_{FS})

_{max}is on the surface. It should be noted that the D

_{FS}distributions for different normal loadings ranging from 0.5P

_{rated}to 1.5P

_{rated}share similar profiles. For this reason, Figure 8a is only shown for a fixed normal load with different μ and μ

_{c}= 0.23 is independent of normal loading.

_{c}associated with (D

_{FS})

_{max}as a function of μ under the rated normal load. For 0 < μ ≤ 0.3, α

_{c}associated with (D

_{FS})

_{max}is not close to 90°. This provides the solid theoretical basis that (D

_{FS})

_{max}can be decreased or (N

_{f})

_{min}can be increased by introducing a proper RS distribution. Since there are four variables to define the RS distribution profile, the optimum RS distribution that can maximize (N

_{f})

_{min}was sought using the optimization scheme introduced in the previous section.

#### 6.2. Increase in (N_{f})_{min} Induced by the Optimum RS Distribution

_{f})

_{min}induced by the optimum RS distribution compared to the case without RS. From the discussion on Figure 8, the D

_{FS}distribution depends on the working condition and so does the optimum RS distribution. Thus, the percentage increase in (N

_{f})

_{min}varies with the working condition. As the normal loading and μ increase, the percentage increase in (N

_{f})

_{min}increases. Compared with the increasing normal loading, μ has a higher impact on the percentage increase in (N

_{f})

_{min}. Moreover, when μ > μ

_{c}, the percentage increase in (N

_{f})

_{min}is dramatically higher than that when μ < μ

_{c}.

_{FS})

_{max}induced by the optimum RS distribution for μ < μ

_{c}. For μ > μ

_{c}, (D

_{FS})

_{max}moves from the surface to the subsurface. In this case, the depth of (D

_{FS})

_{max}when the optimum RS distribution is applied is presented. The increase in the depth of (D

_{FS})

_{max}is beneficial in increasing the propagation life since more loading cycles are needed for the fatigue microcrack to propagate to the surface before a pitting particle is formed [14]. Provided that the relation between the depth of (D

_{FS})

_{max}and the propagation life is quantified, the RS distribution can be optimized to maximize the sum of the initiation life, (N

_{f})

_{min}, and propagation life. Nonetheless, this is out of the scope of the present study.

_{f})

_{min}. Theoretically speaking, the shot peening process parameters can be adjusted to achieve the optimum RS distribution. However, in engineering practice, it is almost impossible to precisely achieve the optimum RS distribution in this way. Thus, it is desired that the optimum RS distribution can be varied to some extent while (N

_{f})

_{min}is not sacrificed too much. By this means, it is more apt to adjust the shot peening process parameters to obtain an acceptable RS distribution and achieve a satisfying (N

_{f})

_{min}. To this end, it is necessary to investigate how the variables of the RS distribution profile affect (N

_{f})

_{min}when they deviate from those of the optimum distribution.

#### 6.3. Effect on (N_{f})_{min} When the RS Distribution Deviates from the Optimum

_{FS}distribution, which has a weak dependence on the normal load P, the effect on (N

_{f})

_{min}when the RS distribution deviates from the optimum is also weakly dependent on P. In light of this, the optimum RS distributions under investigation are those for fixed P with different μ. Figure 10 shows the optimum RS stress distribution for P/P

_{rated}= 1 and μ = 0.1, 0.2, and 0.3. The optimum distributions exhibit two common features. They all have nearly the same (y

_{max})

_{opt}of 0.5b, which is the depth of the maximum Δγ [21], and the same (σ

_{max})

_{opt}= −1000 MPa, which is the lower limit of σ

_{max}. (σ

_{surface})

_{opt}and (y

_{core})

_{opt}do not exhibit a simple dependence on μ since they do not monotonically increase or decrease with μ. The effect on (N

_{f})

_{min}when the RS distribution deviates from the optimum shall be investigated by deviating a single component of (σ

_{surface}, σ

_{max}, y

_{max}, and y

_{core}) from its corresponding optimum value.

_{f})

_{min}when y

_{max}and y

_{core}deviate from their optimum values. As can be seen from Figure 11a, the deviation from (y

_{max})

_{opt}causes a slight decrease in (N

_{f})

_{min}. As μ increases, (N

_{f})

_{min}becomes more sensitive to the deviation of y

_{max}. From Figure 11b, it can be seen that the deviation of y

_{core}almost has no effect on (N

_{f})

_{min}. Figure 12 shows and explains the effect on (N

_{f})

_{min}when σ

_{surface}deviates from (σ

_{surface})

_{opt}. As can be seen from Figure 12a, for relatively low μ, deviation of σ

_{surface}does not affect (N

_{f})

_{min}since (D

_{FS})

_{max}occurs in the subsurface. However, for μ = 0.3, (N

_{f})

_{min}is very sensitive to the deviation of σ

_{surface}unless σ

_{surface}/(σ

_{surface})

_{opt}becomes larger than 0.96. This can be explained by Figure 12b. As |σ

_{surface}| increases, (D

_{FS})

_{max}, which appears on the surface, decreases and the location of (D

_{FS})

_{max}moves from the surface to the subsurface when σ

_{surface}/(σ

_{surface})

_{opt}reaches 0.96. Thereafter, (D

_{FS})

_{max}in the subsurface is not affected by σ

_{surface}and neither is (N

_{f})

_{min}.

_{max})

_{opt}for all the three cases reached its lower limit, −1000 MPa. To investigate the effect on (N

_{f})

_{min}when σ

_{max}deviates from (σ

_{max})

_{opt}, σ

_{max}has to be increased. However, since the constraint in Equation (8b) must be satisfied, increasing σ

_{max}should be coupled with increasing σ

_{surface}. To this end, we simply increased both σ

_{max}and σ

_{surface}by the same amount from (σ

_{max})

_{opt}and (σ

_{surface})

_{opt}, respectively. In this way, the effect of deviation of σ

_{max}was studied. Figure 13 shows (N

_{f})

_{min}and the orientation of the critical plane corresponding to (D

_{FS})

_{max}, α

_{c}, as functions of |σ

_{max}| for four values of μ. The effect of deviation of σ

_{max}on (N

_{f})

_{min}is salient in agreement with the experimental results reported in [45,46,47] that the peak RS plays a major role in the fatigue performance.

_{c}, the location of (D

_{FS})

_{max}remains in the subsurface. As shown in Figure 6 (μ = 0.1), the material point at this location has two critical planes of maximum Δγ, which are denoted as planes 1 and 2 with orientations of α

_{1}and α

_{2}close to 0° and 90°, respectively. Initially, without RS, i.e., |σ

_{max}| = 0, plane 1 has a higher σ

_{n,max}than plane 2 (see Figure 6). According to Equations (5) and (6), the FSDP on plane 1 is larger and is thus equal to (D

_{FS})

_{max}. Consequently, α

_{c}= α. Since the magnitude of the negative normal component of RS on plane 1 is larger than on plane 2 (see Figure 7b), as |σ

_{max}| increases, the FSDP on plane 1 decreases faster and is exceeded by the FSDP on plane 2. As a result, the FSDP on plane 2 becomes (D

_{FS})

_{max}so that α

_{c}switches to α

_{2}. This results in the two distinct stages of (N

_{f})

_{min}vs. |σ

_{max}| in Figure 13a,b.

_{c}, the location of (D

_{FS})

_{max}moves from the surface to the subsurface as |σ

_{max}| increases. As shown in Figure 6 (μ = 0.3), when (D

_{FS})

_{max}is on the surface, the two critical planes of maximum Δγ of the material point at the location of (D

_{FS})

_{max}are denoted as planes 1 and 2. When (D

_{FS})

_{max}moves to the subsurface, the counterparts of planes 1 and 2 are denoted by planes 3 and 4. As can be seen, since α

_{1}≈ 45° and α

_{2}≈ 135°, the negative normal component on planes 1 and 2 are almost the same (see Figure 7b). This results in almost identical FSDP vs. |σ

_{max}| for these two planes. Thus, unlike Figure 13a,b, the switch of α

_{c}from α

_{2}to α

_{1}(see Figure 13d) is not reflected as the deflection on the curve of (N

_{f})

_{min}vs. |σ

_{max}|. Therefore, the two distinct stages of (N

_{f})

_{min}vs. |σ

_{max}| in Figure 13c,d result from the change in the location of (D

_{FS})

_{max}, which is different from the case of μ < μ

_{c}in Figure 13a,b.

_{f})

_{min}vs. |σ

_{max}| being different for μ < μ

_{c}and μ ≥ μ

_{c}, it is interesting to investigate |σ

_{max,t}|, at which the two stages are divided. As demonstrated in Figure 14, |σ

_{max,t}| increases with μ significantly faster when μ > μ

_{c}. The increasing rates of (N

_{f})

_{min}with |σ

_{max}| before and after |σ

_{max,t}| are also presented in this figure. For μ < μ

_{c}, (N

_{f})

_{min}vs. |σ

_{max}| is bilinear and the increasing rate is thus the slope. For μ ≥ μ

_{c}, (N

_{f})

_{min}vs. σ

_{max}is nonlinear when |σ

_{max}| < |σ

_{max,t}|. Thus, the increasing rate is obtained as the average of ∂(N

_{f})

_{min}/∂|σ

_{max}| over 0 < |σ

_{max}| < |σ

_{max,t}| for qualitative analysis. In general, the increasing rate of (N

_{f})

_{min}with |σ

_{max}| when |σ

_{max}| < |σ

_{max,t}| is higher. This indicates that a further increase in |σ

_{max}| over |σ

_{max,t}| becomes less effective in increasing (N

_{f})

_{min}. This theoretical finding is in accordance with experimental results showing that RS induced by additional shot peening is less effective in preventing fatigue in 17NiVrMo6-4 steel [48,49] and Austempered ductile iron [50].

_{f})

_{min}when σ

_{max}deviates from (σ

_{max})

_{opt}. It is true that the ideal scenario is that |σ

_{max}| = |(σ

_{max})

_{opt}| = 1000 MPa. However, a higher |σ

_{max}| requires higher shot velocity that leads to higher surface roughness. Moreover, when |σ

_{max}| > |σ

_{max,t}|, it becomes less effective in increasing (N

_{f})

_{min}by increasing |σ

_{max}|. This is especially problematic for the case of low μ, where α

_{c}is close to 90° (see Figure 13a). In this case, the negative component of RS on this critical plane is trivial and the increasing rate of (N

_{f})

_{min}with |σ

_{max}| is also very slow. Therefore, when one tries to increase (N

_{f})

_{min}by increasing |σ

_{max}|, he may need to comprehensively consider the slowing increasing rate of (N

_{f})

_{min}with |σ

_{max}| and the increasing surface roughness.

## 7. Conclusions

_{c}. The mechanism for RS to increase the fatigue initiation life is that the compressive RS has a negative component on the critical plane and, thus, suppresses the fatigue damage.

_{surface}, y

_{core}, σ

_{max}, and σ

_{surface}. The optimum RS distribution that can increase the fatigue initiation life of the gear surface (N

_{f})

_{min}to the maximum was sought using the GA. The optimum RS distribution is not universal but depends on the working condition. It was demonstrated that the optimum RS distribution can substantially increase (N

_{f})

_{min}under a wide range of working conditions. This improvement in fatigue performance is more significant for higher μ. Another unexpected benefit brought by the optimum RS distribution is that the fatigue initiation occurs at a deeper depth, which increases the fatigue propagation life.

_{f})

_{min}. It was found that for any μ, be it larger or smaller than μ

_{c}, the deviation of y

_{core}and y

_{max}from their optimum values has a negligible effect on (N

_{f})

_{min}. On the other hand, σ

_{max}has a salient effect and |σ

_{max}| should be as large as possible. However, in practice, a cautious decision should be made since increasing |σ

_{max}| increases surface roughness and can have a trivial effect on increasing (N

_{f})

_{min}when μ is low or |σ

_{max}| > |σ

_{max,t}|. The effect of the deviation of σ

_{surface}depends on μ. For μ < μ

_{c}, σ

_{surface}has no effect. For μ > μ

_{c}, σ

_{surface}should be close to (σ

_{surface})

_{opt}, leaving little room for acceptable deviation of σ

_{surface}. This information can serve as a general guidance on RS distribution design by properly modifying the optimum one obtained from GA optimization.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

critical plane | The plane that has the maximum shear strain range over a loading cycle |

FSDP | Fatemi–Socie damage parameter |

RS | Residual stress |

D_{FS} | The maximum FSDP on all the critical planes of a material point |

(D_{FS})_{max} | The maximum D_{FS} of all material points |

N_{f} | Fatigue initiation life of a material point |

(N_{f})_{min} | Minimum fatigue initiation life of all material points or fatigue initiation life of the gear surface |

α | Plane orientation |

α_{c} | The orientation of the critical plane corresponding to D_{FS} |

Δγ | The maximum shear strain range on a plane over a loading cycle |

σ_{n,max} | Maximum normal stress on a plane over a cycle |

σ_{surface} | RS at the surface |

σ_{max} | Peak RS |

y_{max} | Depth of σ_{max} |

y_{core} | Depth where RS vanishes |

σ_{max,t} | The transition value for σ_{max} |

Subscripts | |

opt | Values associated with the optimum RS distribution |

1, 2, 3, … | Values associated with plane 1, 2, 3, … |

## Appendix A

_{m}= (4E

^{*}P/(πR))

^{0.5}is the maximum Hertizian pressure (see Equation (3)) and x′ = (x − x

_{c})/b. In the subsurface,

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**Figure 2.**(

**a**) RS components demonstrated on a gear tooth and (

**b**) a simplified RS distribution profile along the depth.

**Figure 4.**(

**a**) The numerical procedure to determine D

_{FS}and N

_{f}of a material point and (

**b**) the definition of plane orientation α.

**Figure 6.**The contour of Δγ/Δγ

_{max}, σ

_{n,max}/|σ

_{n,max}|

_{max}and the Fatemi–Socie damage parameter (FSDP)/(FSDP)

_{max}for P/P

_{rated}= 1 and μ = 0, 0.1, and 0.3.

**Figure 7.**(

**a**) A typical RS distribution profile and (

**b**) the contour of σ

_{n,r}resulting from this RS distribution.

**Figure 8.**(

**a**) D

_{FS}distribution along the depth for P/P

_{rated}= 1, μ = 0.1, 0.2, and 0.3, and (

**b**) α

_{c}as a function of μ.

**Figure 9.**(

**a**) The percentage increase in (N

_{f})

_{min}, and (

**b**) the percentage increase in the depth of (D

_{FS})

_{max}for μ = 0.1, 0.15, and 0.2 (left vertical axis) and the depth of (D

_{FS})

_{max}for μ = 0.25 and 0.3 (right vertical axis) when the optimum RS distribution is introduced compared to the case without RS.

**Figure 10.**The optimum RS distribution along the depth for P/P

_{rated}= 1 and μ = 0.1, 0.2, and 0.3.

**Figure 11.**The percentage decrease in (N

_{f})

_{min}when (

**a**) y

_{max}and (

**b**) y

_{core}deviate from their optimum values for P/P

_{rated}= 1 and μ = 0.1, 0.2, and 0.3.

**Figure 12.**(

**a**) The percentage decrease in (N

_{f})

_{min}for P/P

_{rated}= 1 and μ = 0.1, 0.2, and 0.3 and (

**b**) the variation of the D

_{FS}distribution along the depth for P/P

_{rated}= 1 and μ = 0.3 when σ

_{surface}deviates from (σ

_{surface})

_{opt}.

**Figure 13.**The effect on (N

_{f})

_{min}and α

_{c}when σ

_{max}deviates from (σ

_{max})

_{opt}= −1000 MPa for P/P

_{rated}= 1 and (

**a**) μ = 0.1, (

**b**) μ = 0.2, (

**c**) μ = 0.25, and (

**d**) μ = 0.3.

**Figure 14.**|σ

_{max,t}| and the increasing rate of (N

_{f})

_{min}with |σ

_{max}| before and after |σ

_{max,t}| as functions of μ.

Parameter | Pinion | Gear |
---|---|---|

Normal module m_{n}/mm | 5.645 | |

Pressure angle α/◦ | 15 | |

Helix angle β/◦ | 22.3 | |

Contact ratio ε_{α} | 1.746 | |

Face width L/mm | 54 | |

Tooth number | z_{1} = 37 | z_{2} = 48 |

Radius at the pitch point/mm | R_{1} = 27.029 | R_{2} = 35.065 |

Rated output torque T_{rated}/(N·m) | 3000 | |

Rotating speed n/(r/min) | 500 | |

Material | AISI 8620RH | |

Young’s modulus/GPa | E_{1} = E_{2} = 210 | |

Poisson’s ratio ν | ν_{1} = ν_{2} = 0.3 | |

Yield strength Y/MPa | 1300 |

Population size, N | 200 |

Maximum generation number, G_{max} | 500 |

Maximum stall generation number, G_{s} | 20 |

Function tolerance, e | 10^{−6} |

Crossover fraction, P_{c} | 0.8 |

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## Share and Cite

**MDPI and ACS Style**

Chen, Z.; Jiang, Y.; Tong, Z.; Tong, S.
Residual Stress Distribution Design for Gear Surfaces Based on Genetic Algorithm Optimization. *Materials* **2021**, *14*, 366.
https://doi.org/10.3390/ma14020366

**AMA Style**

Chen Z, Jiang Y, Tong Z, Tong S.
Residual Stress Distribution Design for Gear Surfaces Based on Genetic Algorithm Optimization. *Materials*. 2021; 14(2):366.
https://doi.org/10.3390/ma14020366

**Chicago/Turabian Style**

Chen, Zhou, Yibo Jiang, Zheming Tong, and Shuiguang Tong.
2021. "Residual Stress Distribution Design for Gear Surfaces Based on Genetic Algorithm Optimization" *Materials* 14, no. 2: 366.
https://doi.org/10.3390/ma14020366